发表机构
Univ. de Sevilla（塞维利亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在一般测度空间上研究非局部半线性扩散方程，通过核诱导连通结构完整刻画强极值原理，发展上下解方法，获得解的存在性、唯一性、不存在性及正则性结果，并给出数值实验验证。
AI 中文摘要
我们研究一般测度空间$(\Omega,M,\mu)$上如下形式的非局部半线性扩散方程的非平凡解的存在性：\\[ a_0(x)u(x)-\int_\Omega k(x,y)\big(u(y)-u(x)\big)\\,d\mu(y)=F(x,u), \quad u\geq 0, \\] 其中核$k$是非负且对称的。在此框架下，一个主要挑战是建立解的严格正性。为此，我们基于核诱导的连通结构，给出了强极值原理的一个新的完整刻画，该刻画导致定义域分解为独立分量。我们还在此设定下发展了上下解方法，获得了存在性、唯一性和不存在性结果，以及在适当假设下保证解连续性的正则性结果。在抽象测度空间中工作的主要兴趣在于其丰富的适用性，因为它统一并推广了多种框架，包括离散、连续和混合模型，或跨不同维度相互作用的系统。最后，我们给出数值实验以说明理论结果的潜力。
英文摘要
We study the existence of nontrivial solutions to nonlocal semilinear diffusion equations of the form \[ a_0(x)u(x)-\int_Ωk(x,y)\big(u(y)-u(x)\big)\,dμ(y)=F(x,u), \quad u\geq 0, \] on a general measure space $(Ω,M,μ)$, where the kernel $k$ is nonnegative and symmetric. In this framework, a major challenge is establishing the strict positivity of solutions. To this end, we provide a novel complete characterization of the strong maximum principle in terms of a connectivity structure induced by the kernel, which leads to a decomposition of the domain into independent components. We also develop the method of sub- and supersolutions in this setting, obtaining existence, uniqueness, and nonexistence results, as well as a regularity result that ensures the continuity of solutions under suitable assumptions. The main interest of working within an abstract measure space lies in its rich applicability, as it unifies and extends a wide variety of frameworks, including discrete, continuous, and hybrid models, or systems interacting across different dimensions. Finally, numerical experiments are presented to illustrate the potential of our theoretical results.
